# THE LAWS OF MOTION

## Mass :-

This is the quantity to quantify the amount of matter present in an object. It is a scalar quantity.

## Inertia :-

This is the tendency of an object to resist its state of motion, rest or direction. For instance, when an object is on the bus, we tie it with a rope. This is because when the bus suddenly starts, due to the resistance of state of the object, it tends to fall down as it wants to be at rest. Also, if the bus suddenly stops, it can fall because it wants to be in motion.

## FORCE

It is a push or pull acted on a body. It is a vector quantity because we always apply force in a particular direction. The SI unit of force is Newton. The reason for the SI unit will be stated in the explanation of the second law of motion.

## Momentum :-

It is mass in motion. This means If I apply some force on an object, the object gains velocity and hence the object possesses some amount of motion. This motion is called momentum.

### Example:-

1.- Let us say that there is a man between two mountains on a road and a cycle and a truck are coming from 2 points by sliding down the mountain. Now, let me ask you a question. Which side would you go to push any one of the objects? Opposite to the cycle or the truck? Why? The answer to this question is you would go opposite to the cycle and push it because the mass of it is very less and when it was coming down, it would have a lesser velocity than that of the truck and hence you can save yourself. So, when I told you this example, I want you to infer that momentum depends on mass and velocity and the mass of the object possesses certain velocity and moves with it. This phenomenon is called momentum

In a mathematical way, we say that momentum is the product of mass and velocity. Also, momentum is denoted by the letter P. We denote it with letter P because the letter m is already used for mass.

The equation is: -

P = mv

Momentum is a vector quantity because it has a quantity called velocity which is a vector quantity. A vector quantity is a quantity which depends on magnitude, direction, vector laws of addition and vector algebra.

## First Law of Motion :-

The first Law of Motion says that a body will remain at rest until and unless and until an unless a net external force acts on the body.

The first Law of Motion also says that a body will remain in constant velocity until and unless a net external force acts on the body.

This means that if a spaceship is launched and it reaches the space, away from any any external force like gravity and then it's engine has turned on, then it will continue to go at that velocity even when the engine is turned off.

## Second Law of Motion :-

The second law of motion says that the net external force acting on a body is directly proportional to the Rate of Change of Momentum. Also, the mass is inversely proportional to the acceleration because of inertia. More the mass, less it will accelerate.

The below explanation is the derivation of formula of second law of motion.

F **∝ Rate of change of Momentum**

**F****∝ (****Mv-Mu)/t**

**F****∝ M****(v-u)/t [(Final Velocity- Initial Velocity)/t=a] **

**F****∝ Ma**

**F=k ma [ k=1 because 1 Newton=1 kg-m/s**² **]**

**F=ma **

**NOTE :-** F** **≠ ma because the second law says that the net force is equal to ma.

**F= Net External Force Acting on The Body**

**v= Final Velocity**

**u= Initial Velocity**

**M=Mass**

## THE THIRD LAW OF MOTION

The Third Law of Motion states that every action has an equal and opposite reaction. An important thing to take note of is that when two bodies are moving with the same velocity, the body with the lesser mass will displace more than the body with the greater mass. This is because : -

1) From the second law of motion, we know that the mass is inversely proportional to acceleration.

2) Hence, the object with lesser mass will displace more than the object with greater mass.

NOTE- It always acts on two different bodies.

## Conservation of Momentum :-

We know that when we apply force to an object, it is directly proportional to the rate of change of momentum which is the second law of motion. Now, what if we do not apply force to any 2 objects (which are a part of our system) and they move with a certain momentum such that they collide after a while. Then, we can observe that there is no force and the bodies are moving with some initial momentum and they will collide at some point. Now, the law of conservation of momentum states that when there is no force acting on the object, then the total initial momentum before collision of the objects is equal to the total final momentum after collision of the objects.

Now, let us see how to derive this formula

When we derive this formula, we use the following laws of motion: -

1) Second Law

2) Third Law

You can use the picture at the last to look at the derivation of the formula. Let us say that the masses of the objects are m1 and m2 respectively. And the initial velocities of both the objects are u1 and u2 respectively. Now, let us say there is a force F1 acting on the body of mass m1. Now, the velocity of the mass m1 will change after the action of force to v1 and after sometime it will collide with the second object with mass m2.

Now, when the objects collide, then the mass m2 will act a force F2 on the mass m1 which equal and opposite. This is from the third law of motion. Now, the velocity of the body of mass m2 will also change as there is application of force to v2. Now, if these concepts are clear, then we can proceed to the derivation of this formula.

Please read through this information while looking at the derivation

**Q1. There are cars with masses 4 kg and 10 kg respectively that are at rest. A car having the mass 10 kg moves towards the east with a velocity of 5 m.s**^{-1}**. Find the velocity of the car with mass 4 kg with respect to ground.**

**Ans:** Given,

m_{1} = 4 kg

m_{2} = 10 kg

v_{1} = ?

v_{2} = 5 m.s^{-1}

We know from the law of conservation of momentum that,

P_{initial} = 0, as the cars are at rest

P_{final} = p_{1} + p_{2}

P_{final} = m_{1}.v_{1} + m_{2}.v_{2}

= 4 kg.v_{1} + 10 kg.5 m.s^{-1}

P_{i} = P_{f}

0=4 kg.v_{1}+50 kg.m.s^{-1}

v_{1} = 12.5 m.s^{-1}

You can read through one of the numerical examples

I will give you a practice question. Please try to solve it

A man weighing 60 kg runs along the rails with a velocity of 18 km/h and jumps into a car of mass 1 quintal (100 kg) standing on the rails. Calculate the velocity with which car will start travelling along the rails.

Derivation of conservation of momentum is the above picture attached. Please refer to it to understand better. Also, read through the information above while walking through this picture.

### Slight Changes In This Formula

Their are many types of collisions and because so many types of collisions, this formula is changed.

The changes are-

1) When The Bodies collide and stick with each other, they will have a common mass and also a common velocity- Final momentum= (m1+m2)V

** The Formula will become=m**_{1}**u**_{1}**+m**_{2}**u**_{2}**= (m**_{1}**+m**_{2}**)V**